#500811

order	512
nilpotency class	4
isoclinism	isoclinic class of #500807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24260
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3018	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

999 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 138 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,18,42)(4,9,22,36)(5,23,14,10)(6,25,28,11)(8,29,32,16)(15,40,47,63)(17,49,26,41)(19,51,50,64)(20,53,43,34)(21,35,27,39)(24,37,44,55)(30,57,59,46)(31,48,38,56)(33,61,60,52)(45,62,54,58);(1,3,14,29,7,18,5,16)(2,8,23,42,12,32,10,13)(4,19,43,61,22,50,20,52)(6,26,44,48,28,17,24,56)(9,33,53,64,36,60,34,51)(11,38,55,41,25,31,37,49)(15,45,57,27,47,54,46,21)(30,58,63,39,59,62,40,35);(1,4)(2,9)(3,15)(5,20)(6,27)(7,22)(8,30)(10,34)(11,39)(12,36)(13,40)(14,43)(16,46)(17,50)(18,47)(19,26)(21,28)(23,53)(24,45)(25,35)(29,57)(31,60)(32,59)(33,38)(37,58)(41,64)(42,63)(44,54)(48,61)(49,51)(52,56)(55,62);(1,3,14,29,7,18,5,16)(2,8,23,42,12,32,10,13)(4,15,43,57,22,47,20,46)(6,17,44,56,28,26,24,48)(9,30,53,63,36,59,34,40)(11,31,55,49,25,38,37,41)(19,45,61,27,50,54,52,21)(33,58,64,39,60,62,51,35)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341810, #72341964, #72341976, …
maximal normal subgroup of	#72341810, #72341976, #72342006, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408007);